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A real number as a point

  1. Jul 15, 2015 #1
    An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) ##p## in the domain of a real-valued function ##f## of a real variable where ##f'(p)=0## or ##f'(p)## is undefined is called a critical point of the function. The particular type of critical point ##x## where ##f'(x)=0## is called a stationary point. As another example, "the graph ##y=(x+2)(x-1)^2## cross the x-axis at the points ##x=-2## and ##x=1##." In those cases, why we called a real number as a point? Is it because we view the real numbers as points in the context of the real line?
     
  2. jcsd
  3. Jul 15, 2015 #2
    On the graph I think I should say it cross ##x##-axis at the points ##(-2,0)## and ##(1,0).## I think this is more specfic, but I'm not sure whether your statement is wrong or not.
    I think your former example's reason is that the whole ##x-y## plane is the domain of the function. They are also true points, aren't they?
     
  4. Jul 16, 2015 #3

    Mark44

    Staff: Mentor

    Each point on a number line represents a real number.

    In shuxue's example, it was clearly stated that the points were on the x-axis, so the point on the x-axis where x = -2 is also the ordered pair (-2, 0).
    No, in the first example, it says that the function is of one variable, so the domain is some subset of the real numbers, possibly including the entire real number line.
    ???
    I don't understand what this refers to.
     
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